An “offline” edit is an experimental/rough cut, usually done on cheap or even domestic equipment in order to decide on the final edit decision list (EDL) – a list of each shot in the final movie. Once the exact in/out points have been determined the EDL exported and edit finalised in the “online” edit suite with high-end professional equipment.
This whole process is dictated by cost: the offline edit takes longer (due to experimentation), so it is done on the cheap; the online edit can be done much faster using the previously created EDL, thus minimising time (which is money) in the professional editing suite.
As computer power has grown (and prices fallen), the line between what is constitutes “online” and “offline” equipment is becoming increasingly blurred, certainly in the editing arena (high-quality visual effects tend to still be more the domain of expensive equipment). It’s possible to do an “online” edit on Final Cut Pro or Avid Xpress Pro if your production dictates it. Likewise you may decide to do an offline on one of these packages, and use the resulting EDL to work on higher-end Avids or even a Final Cut Pro suite which has expensive hardware components.